### Doodles

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

### Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

### N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

# Counting Binary Ops

##### Stage: 4 Challenge Level:

This question is about the number of ways of combining an ordered list of terms by repeating a single binary operation.

For example with three terms $a$, $b$ and $c$ there are just two ways $((a\oplus b)\oplus c)$ and $(a\oplus (b\oplus c))$. Suppose the binary operation $\oplus$ is just ordinary subtraction and $a=12, \; b=7, \; c=5$ then $((a\oplus b)\oplus c)= 5 - 5 =0$ and $(a\oplus (b\oplus c))= 12 - 2 = 10$.

We are not concerned in this question with doing the 'arithmetic' or with whether the answers are the same or different. We just want to find out how many ways there are of combining the terms, or if you like of putting brackets into the expression. Note that we need the brackets because the answers may be different as in the subtraction example.

These two tree diagrams show the two cases for combining 3 terms."

Show that for four terms, (three binary operations) there are five cases and find the number of cases for five terms and six terms.